How do you scale eye texture using armatures?
You may do it using drivers. You can set the shapekey's transformation to be driven by the bone's scale as pictured below.

First select the eyeball and add a new shapekey being in Object Mode. Next enter the Edit Mode, set the shapekey's value to 1.000 (with the apply shapekeys in edit mode button enabled) and edit your mesh as you like (in my case I've decreased the pupil's size).

You may set the Min shapekey's value to -1.000, so Blender may create the opposite transformations (pupil's size enlargement) automatically for you

Now being in Object Mode add the single bone and place it in front of the eyeball (name it e.g. eye_pupil).

Now select the eyeball, go to the Shape Keys panel, hover the cursor over the newly created shapekey's value (near it's name), press RMB and choose Add Driver.

Next open the Graph Editor and set it's mode to Drivers. Select the shapekey's name on the left, open the properties shelf (N) and set the settings there as pictured below (you want the shapekey value to be driven by the local Scale value of the previously created eye_pupil bone using the Transform Channel).

You may also add a generator modifier to set the ranges of the driven values (but it's not neccessary).

Finally you'll have to update dependencies (just click the button pictured below).

NOTE: My answer shows how to animate the pupil only as I have the ready model which uses this kind of animation only. The video you provided shows the animation of the iris with pupil, but with the method pictured above you'll be able to do it in the same manner.